Generalization · integer p ≥ 2 · positive X

A fixed line for every pth root

The same triangle chain and green parallel. The correction is transferred to AK, with x_K/CB = 1 − X/p. The geometry implements Newton in the internal variable 1/s.

Degree p Target X Internal state s :

All three parameters immediately recompute the construction. The proof covers every integer p≥2 and all X,s>0; the controls above only limit the display.

Readout L_p−1B_p−1 = Xs^p−1

Next readout

Gray: rectangle and diagonal. Blue: horizontals. Red: telescoping parallels. Green: ME and correction. Purple: AK. Extensions are shown as needed. Only selected points are labeled at large degrees.

Formula derived from intersections

s⁺ = ps / (p − 1 + Xsᵖ).

Global quadratic convergence for every positive starting point. For X>1, the original construction is linear near the solution. This formula is Newton after inversion; it is not numerically novel. Historical priority for this exact drawing remains undetermined.